Tangential quadrilateral

An example of a tangential quadrilateral

In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides are all tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since tangential quadrilaterals can be drawn surrounding or circumscribing their incircles, they are also sometimes called circumscribable quadrilaterals. Another name for the same class of quadrilaterals is inscriptable quadrilaterals.

Special cases

Examples of tangential quadrilaterals are squares, rhombi, and kites. The kites are exactly the tangential quadrilaterals that are also orthodiagonal. If a quadrilateral is both tangential and cyclic, it is called a bicentric quadrilateral.

Characterizations

Chao and Simeonov's characterization in terms of the radii of circles within each of four triangles

In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle. Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter.

According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add to the same total length, which equals the semiperimeter of the quadrilateral:

$a + c = b + d = \frac{a + b + c + d}{2} = s.$

Conversely a convex quadrilateral in which a + c = b + d must be tangential.

If opposite sides in a convex quadrilateral ABCD (that is not a trapezoid) intersect at E and F, then it is tangential if and only if either of

$\displaystyle BE+BF=DE+DF$

or

$\displaystyle AE-EC=AF-FC.$

The second of these is almost the same as one of the equalities in Urquhart's Theorem. The only differences are the signs on both sides; in Urquhart's Theorem there are sums instead of differences.

Another necessary and sufficient condition is that a convex quadrilateral ABCD is tangential if and only if the incircles in the two triangles ABC and ADC are tangent to each other.

Chao and Simeonov observed another characterization of tangential quadrilaterals. The two diagonals of any convex quadrilateral partition the quadrilateral into four triangles. Let r₁, r₂, r₃, and r₄ denote the radii of the circles inscribed in the four successively adjacent triangles; then the quadrilateral is tangential if and only if

$\frac{1}{r_1}+\frac{1}{r_3}=\frac{1}{r_2}+\frac{1}{r_4}.$

This characterization had previously been proved by Vaynshtejn in 1995. In the solution to his problem, another similar characterization was given by Vasilyev and Senderov. If h₁, h₂, h₃, and h₄ are the altitudes in the same four triangles (from the diagonal intersection to the sides of the quadrilateral), then the quadrilateral is tangential if and only if

$\frac{1}{h_1}+\frac{1}{h_3}=\frac{1}{h_2}+\frac{1}{h_4}.$

Yet another similar characterization concerns the exradii R₁, R₂, R₃, and R₄ in the same four triangles (the four excircles are each tangent to one side of the quadrilateral and the extensions of its diagonals). The quadrilateral is tangential if and only if

$\frac{1}{R_1}+\frac{1}{R_3}=\frac{1}{R_2}+\frac{1}{R_4},$

that is, the sums of opposite inverse exradii are equal.

In 1996, Vaynshtejn was probably the first to prove another nice characterization of tangential quadrilaterals, that has later appeared in several magazines and websites. It states that when a convex quadrilateral is divided into four nonoverlapping triangles by its two diagonals, then the incenters of the four triangles are concyclic if and only if the quadrilateral is tangential. In fact, the incenters form a cyclic orthodiagonal quadrilateral. A similar variant is that the incircles can be exchanged for the excircles to the same triangles (tangent to the sides of the quadrilateral and the extensions of its diagonals). Thus a convex quadrilateral is tangential if and only if the excenters in these four excircles are the vertices of a cyclic quadrilateral.

A characterization regarding the angles formed by diagonal BD and the four sides of a quadrilateral ABCD is due to Iosifescu. He proved in 1954 that a convex quadrilateral has an incircle if and only if

$\tan{\frac{\angle ABD}{2}}\cdot\tan{\frac{\angle BDC}{2}}=\tan{\frac{\angle ADB}{2}}\cdot\tan{\frac{\angle DBC}{2}}.$

Area

Tangential quadrilateral with inradius r.

The area K of a tangential quadrilateral is

$\displaystyle K = r \cdot s,$

where s is the semiperimeter and r is the inradius.

A trigonometric formula for the area is

$\displaystyle K = \sqrt{abcd} \sin \frac{B+D}{2}$

where B and D are opposite angles. For given side lengths, the area is maximum when the quadrilateral is also cyclic and hence a bicentric quadrilateral. Then $K = \sqrt{abcd}$ since opposite angles are supplementary angles. This can be proved in another way using calculus.

In fact, the area can be expressed in terms of just two adjacent sides and two opposite angles as

$K=ab\sin{\frac{B}{2}}\csc{\frac{D}{2}}\sin \frac{B+D}{2}.$

Another area formula is

$K=\tfrac{1}{2}(ac-bd)\tan{\theta},$

where θ is the angle between the diagonals. Still another is

$\displaystyle K = \tfrac{1}{2}\sqrt{p^2q^2-(ac-bd)^2}$

which gives the area in terms of the diagonals p, q and the sides a, b, c, d of the quadrilateral.

The area K can also be expressed in terms of the four tangent lengths. (The tangent length from any vertex is the distance from that vertex to either of the points where the incircle is tangent to the sides emanating from that vertex.) If these are e, f, g, h, then the tangential quadrilateral has the area

$\displaystyle K=\sqrt{(e+f+g+h)(efg+fgh+ghe+hef)}.$

Inradius

The inradius in a tangential quadrilateral with consecutive sides a, b, c, d is given by

$r=\frac{K}{s}=\frac{K}{a+c}=\frac{K}{b+d}$

where K is the area of the quadrilateral. For a tangential quadrilateral with given sides, the inradius is maximum when the quadrilateral is also cyclic (and hence a bicentric quadrilateral).

In terms of the tangent lengths, the incircle has radius

$\displaystyle r=\sqrt{\frac{efg+fgh+ghe+hef}{e+f+g+h}}.$

The inradius can also be expressed in terms of the distances from the incenter I to the vertices of the tangential quadrilateral ABCD. If u = AI, v = BI, x = CI and y = DI, then

$r=2\sqrt{\frac{(\sigma-uvx)(\sigma-vxy)(\sigma-xyu)(\sigma-yuv)}{uvxy(uv+xy)(ux+vy)(uy+vx)}}$

where $\sigma=\tfrac{1}{2}(uvx+vxy+xyu+yuv)$.

Diagonals and tangency chords

If e, f, g and h are the tangent lengths from A, B, C and D respectively to the points where the incircle is tangent to the sides of a tangential quadrilateral ABCD, then the lengths of the diagonals p = AC and q = BD are

$\displaystyle p=\sqrt{\frac{e+g}{f+h}\Big((e+g)(f+h)+4fh\Big)},$

$\displaystyle q=\sqrt{\frac{f+h}{e+g}\Big((e+g)(f+h)+4eg\Big)}.$

The two line segmets that connect the points of tangency of the incircle at opposite sides of the tangential quadrilateral are called the tangency chords. The lengths k and l of these can also be expressed in terms of the tangent lengths as

$\displaystyle k=\frac{2(efg+fgh+ghe+hef)}{\sqrt{(e+f)(g+h)(e+g)(f+h)}},$

$\displaystyle l=\frac{2(efg+fgh+ghe+hef)}{\sqrt{(e+h)(f+g)(e+g)(f+h)}}.$

The two diagonals and the two tangency chords are concurrent. One way to see this is as a limiting case of Brianchon's theorem, which states that a hexagon all of whose sides are tangent to a single conic section has three diagonals that meet at a point. From a tangential quadrilateral, one can form a hexagon with two 180° angles, by placing two new vertices at two opposite points of tangency; all six of the sides of this hexagon lie on lines tangent to the inscribed circle, so its diagonals meet at a point. But two of these diagonals are the same as the diagonals of the tangential quadrilateral, and the third diagonal of the hexagon is the line through two opposite points of tangency. Repeating this same argument with the other two points of tangency completes the proof of the result.

Properties of the incenter

If a line cuts a tangential quadrilateral into two polygons with equal areas and equal perimeters, then that line passes through the incenter.

If I is the incenter of a tangential quadrilateral ABCD, then

$IA\cdot IC+IB\cdot ID=\sqrt{AB\cdot BC\cdot CD\cdot DA}.$

The incenter I in a tangential quadrilateral ABCD coincides with the centroid of the quadrilateral if and only if

$IA\cdot IC=IB\cdot ID.$

If M and N are the midpoints of the diagonals AC and BD respectively in a tangential quadrilateral ABCD with incenter I, then the points M, I, and N are collinear. This line is called the Newton line of the quadrilateral. Also,

$\frac{IM}{IN}=\frac{IA\cdot IC}{IB\cdot ID}=\frac{e+g}{f+h}$

where e, f, g and h are the tangent lengths at A, B, C and D respectively. Combining the first equality with a previous property, the centroid of the tangential quadrilateral coincides with the incenter if and only if the incenter is the midpoint of the line segment connecting the midpoints of the diagonals.

Other properties

• The four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kites.

• If a four-bar linkage is made in the form of a tangential quadrilateral, then it will remain tangential no matter how the linkage is flexed, provided the quadrilateral remains convex.

• If the incircle is tangent to the sides AB, BC, CD, DA at W, X, Y, Z respectively, then the lines WX, ZY and AC are concurrent.

Conditions for a tangential quadrilateral to be another type of quadrilateral

A tangential quadrilateral is a rhombus if and only if its opposite angles are equal.

If the incircle is tangent to the quadrilateral at points W, X, Y, Z in sequence, then the tangential quadrilateral is also cyclic (and hence bicentric) if and only if WY is perpendicular to XZ. The quadrilateral WXYZ is called the contact quadrilateral.

A tangential quadrilateral is a kite if and only if any one of the following conditions is true:

• The area is one half the product of the diagonals
• The diagonals are perpendicular
• The two line segments connecting opposite points of tangency have equal length
• One pair of opposite tangent lengths have equal length
• The bimedians have equal length
• The products of opposite sides are equal
• The center of the incircle lies on the longest diagonal

See also

• Bicentric quadrilateral
• Cyclic quadrilateral
• Ex-tangential quadrilateral
• Kite
• Orthodiagonal quadrilateral
• Rhombus

External links

• Weisstein, Eric W., "Tangential Quadrilateral" from MathWorld.